Accusation breeds guilt | OUPblog
The following are two simple results that constraint the number of liars, and the number of truth-tellers, in any such group (I’ll provide proofs of these results in the comments after a few days).
“Accusations of untrustworthiness tend to decrease the overall level of trust we place in those involved”
Result 1: If, for some number m, each person in the group accuses at least m other people in the group of lying (and there is no paradox) then there are at least m liars in the group.
Result 2: If, for any two people in the group p1 and p2, either p1 accuses p2 of lying, or p2 accuses p1 of lying (and there is no paradox), then exactly one person in the group is telling the truth, and everyone else is lying.
These results support an affirmative answer to our question: Given a group of people, the more accusations of untrustworthiness (i.e., of lying) are made, the higher the minimum number of people in the group that must be lying. If there are enough accusations to guarantee that each person accuses at least n people, then there are at least n liars, and if there are enough to guarantee that there is an accusation between each pair of people, then all but one person is lying. (Exercise for the reader: show that there is no situation of this sort where everyone is lying).